Tensor-train approximation of the chemical master equation and its application for parameter inference

Area of Science:

• Computational Chemistry
• Applied Mathematics
• Chemical Kinetics

Background:

• The chemical master equation (CME) models stochastic chemical reactions but becomes computationally intensive in high dimensions.
• Existing methods struggle with the curse of dimensionality and incorporating parametric dependencies efficiently.
• Representing the probability mass function (PMF) of CME solutions is crucial for accurate inference.

Purpose of the Study:

• To develop and apply a novel tensor-train (TT) format for Bayesian inference tasks related to the CME.
• To leverage the TT format for efficient representation and computation of high-dimensional CME solutions.
• To demonstrate the incorporation of parametric dependencies within the TT framework for enhanced modeling.

Main Methods:

• The study employs the tensor-train approximation to represent the PMF of the CME solution.
• Time is integrated as an additional dimension within the tensor structure, forming a linear system for temporal evolution.
• Parametric dependencies are incorporated using a tensor product basis expansion in the parameter space.

Main Results:

• The TT format achieves a very high compression ratio for storing the CME solution's PMF.
• Significant reductions in computational time are observed due to linear algebra operations performed in the TT format.
• Successful execution of inference tasks, including smoothing and parameter inference, using the TT framework.

Conclusions:

• The tensor-train format offers an efficient and computationally advantageous approach for solving the chemical master equation.
• This method effectively handles high-dimensional problems and facilitates the incorporation of parametric dependencies.
• The TT-based Bayesian inference framework shows promise for accelerating and improving the accuracy of chemical reaction modeling.